Stability theory of solutions of dynamic equations on time scales (Q2915712)

This monograph is the first book in the mathematical literature completely devoted to stability theory for dynamic equations on time scales. The book is innovative and systematically developed. It has been written by one of the well-known experts in the theory of stability, in particular, Lyapunov's direct method. The presentation of the material itself is detailed, all new results are supplied with complete proofs, many examples and enlightening explanations. Besides that, each chapter is accompanied by notes on the locally related bibliography. Throughout the book, an imposing collection of particular stability results is presented, among them a large number of new results by the author.NEWLINENEWLINEThe book consists of five chapters. Chapter 1 contains a clear description of the main concepts, definitions, and theorems from mathematical analysis on time scales (without proofs). In Chapter 2, the reader finds some results for dynamic integral inequalities, starting from inequalities of Gronwall-Bellman-type. Their application to problems of stability of solutions of linear, quasi-linear and nonlinear dynamic equations forms the main content of this chapter. Chapter 3 contains a generalization of Lyapunov's direct method for dynamic equations. The main theorems of the method are presented, including theorems on exponential stability and polystabllity on time scales. The construction of appropriate Lyapunov functions is based on auxiliary matrix-valued functions. A special section of the chapter is dedicated to the inversion of the theorem on exponential stability. A method of construction of Lyapunov functions as a quadratic form for linear dynamic equations is presented as well. Chapter 4 contains the development of the comparison method for dynamic equations on time scales. For dynamic equations, the comparison method is presented in the context with an auxiliary matrix-valued function and dynamic comparison equations. On the basis of general theorems of the comparison principle, some problems of stability of invariant sets, the stability with respect to two measures and the practical stability are discussed here. In the closing chapter, Chapter 5, some general results of the previous chapters are applied to dynamic equations having real physical sense both in the continuous and in the discrete cases. These are problems of stability of dynamic neuron networks, population dynamics, Ramsey models in economics, and dynamic equations with structural perturbations.NEWLINENEWLINEThe bibliography of about 100 titles obviously does not claim completeness. The book can be warmly recommended to everybody interested in the subject. It will be useful to specialists in dynamical systems, applied mathematics, mechanics and graduate students studying differential, difference equations and control theory.